1. ## Basis

Show that B = {x^2, (x -1)^2, (x + 1)^2}is a basis for P2

I'm not even sure how to do this, so any help would be great!

2. Well, you want to show that any polynomial in $P_{2}$ can be written as a linear combination of elements of $B.$ That is, for an arbitrary element $ax^{2}+bx+c\in P_{2},$ you want to show that

$ax^{2}+bx+c=tx^{2}+u(x-1)^{2}+v(x+1)^{2},$

where $t,u,v$ are constants to be determined in terms of $a,b,c.$ Does that give you some direction?

3. That helps somewhat, but I'm still abit confused. So am I trying to find what t, u, and v are?

As hint we were told to try and evaluate a linear combination at -1, 0, and 1. How would I do this?

4. Originally Posted by skittle
That helps somewhat, but I'm still abit confused. So am I trying to find what t, u, and v are?

As hint we were told to try and evaluate a linear combination at -1, 0, and 1. How would I do this?
Well, the equation exhibited in post # 2 has to hold for all values of x. If you plug in those three values, one at a time, you get three fairly simple equations for the three unknowns. Then you can solve for them. And, assuming you get even one solution, you will have proven what you need to prove.

5. Okay, I think I got it now. Thanks so much for the help!

6. You're welcome!

7. Sorry to bug you again, but I just I was having trouble evaluating the equation at 1 and -1. Could you explain to me how to do this?

8. I'll do one for you: x = 1:

$a(1)^{2}+b(1)+c=t(1)^{2}+u(1-1)^{2}+v(1+1)^{2}$

$a+b+c=t+4v.$

9. I worked out that

x = 0 is c = u+v x = 1 is a+b+c = t + 4v x = -1 is a-b+c = t + 4u

What do I do from here?

10. You have three equations in the three unknowns t, u, v. Solve for them. If you find at least one solution, you're essentially done.

11. Could you show me an example of how to solve for one of them?

Thanks

12. Substitution, or elimination is probably the best way to go.

13. I've been trying the methods you suggested to solve for the unknowns, but I have been trouble with it. I was wondering if you could demonstrate how to do it?

14. That's not generally the way this forum works. Why don't you post your work, and show me exactly where you're having trouble, and I'll help you get unstuck?

15. I get here and then get stuck:

a+b+c= t+4v
a-b+c= t+4u
c = u+v

I work it out to 2a+3c = 2t +5u+5v

And I'm not sure what to do from here.

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